Uniform Boundedness of a Preconditioned Normal Matrix Used in Interior-Point Methods

Uniform Boundedness of a Preconditioned Normal Matrix Used in Interior-Point Methods

Systems of linear equations with “normal” matrices of the form ADA is a key ingredient in the computation of search directions for interior-point algorithms. In this article, we establish that a well-known basis preconditioner for such systems of linear equations produces scaled matrices with uniformly bounded condition numbers as D varies over the set of all positive diagonal matrices. In part...

Uniform boundedness of the inverse of a Jacobian matrix arising in regularized interior-point methods

This short communication analyses a boundedness property of the inverse of a Jacobian matrix that arises in regularized primal-dual interior-point methods for linear and nonlinear programming. This result should be a useful tool for the convergence analysis of these kinds of methods.

Sparse Matrix Ordering Methods for Interior Point

The main cost of solving a linear programming problem using an interior point method is usually the cost of solving a series of sparse, symmetric linear systems of equations, AA T x = b. These systems are typically solved using a sparse direct method. The rst step in such a method is a reordering of the rows and columns of the matrix to reduce ll in the factor and/or reduce the required work. T...

Interior-point Methods

Final Iterations in Interior Point Methods | Preconditioned Conjugate Gradients and Modiied Search Directions

In this article we consider modiied search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modiied search directions are computed by solving perturbed systems in which the systems may be solved ef-ciently by the preconditioned conjugate gradient solver. We prove the conv...